Can a system of cubic anharmonic oscillators have multiple stable equilibria?

Generically, I would bet on the scenario that there is always only one stable equilibrium and a bunch of unstable ones.

Your Hamiltonian looks like $$H = frac{1}{2m_1}, p_1^2 + .. + frac{1}{2m_n}, p_n^2 + U^{(3)}(q_1, ..., q_n)$$ where $U^{(3)}$ is a sum of a homogeneous quadratic plus a homogeneous cubic cubic polynomial with respect to the variables $q_1,...,q_n$. The equilibrium points are the solutions to the algebraic (no more than inhomogeneous quadratic) equations: begin{align} &frac{partial H}{partial p_1} = frac{1}{m_1}, p_1 = 0 &... &frac{partial H}{partial p_n} = frac{1}{m_1}, p_2 = 0 &frac{partial H}{partial q_1} = frac{partial U^{(3)}}{partial q_1}(q_1, ..., q_n) = 0 &... &frac{partial H}{partial q_n} = frac{partial U^{(3)}}{partial q_n}(q_1, ..., q_n) = 0 end{align} Since the first half of the equations yields $p_1 = ... = p_n = 0$, you are left with the quadratic equations from the second half: begin{align} &frac{partial H}{partial q_1} = frac{partial U^{(3)}}{partial q_1}(q_1, ..., q_n) = 0 &... &frac{partial H}{partial q_n} = frac{partial U^{(3)}}{partial q_n}(q_1, ..., q_n) = 0 end{align} only for the variables $q_1, ..., q_n$. Thus, since $H$ can serve as a Lyapunov stability function, the local minima of the cubic multi-variable polynomial are the stable (but not asymptotically) equilibria. You can check that since in this special case $U^{(3)}$ is a sum of a homogeneous quadratic and a homogeneous cubic polynomial, the point $q_1 = ... = q_n = 0$ is always a stable equilibrium, because $U^{(3)}$ has a local minimum there, assuming that the parameters $k_i$ are all positive, which should be the case of harmonic oscilators. However, if you pick another equilibrium of $H$, call it equilibrium 1, then noting stops you from drawing a one-dimensional line in the space $q_1,...,q_n$ that connects the zero equilibrium to equilibrium 1. Furthermore, if you restrict the polynomial $U^{(3)}$ onto this line, you obtain a single variable cubic polynomial that has a local minimum at the zero equilibrium. Then the other equilibrium 1, is also a critical point and the only option is for it to be a local maximum for $U^{(3)}$ restricted to the line, which means that equilibrium 1 cannot be a local minimum in the ambient space $q_1,...,q_n$ and hence, cannot be a stable equilibrium.